I’ve played with R a bit more, and figured out how to find intersections and draw overlap graphs, like this:
You can work out the probability that the number 1 song has less votes than this (25.39%) and the probability that song number 2 is more than this (24.16%). Then, if my reasoning is correct, you can figure out the probability that song 1 is lower than this and that song 2 is higher than this by multiplying these two probabilities: 6.14%.
Now, I think this is the probability that songs 1 and 2 will swap places in the actual countdown, but I’m not sure. It’s possible that I should actually be calculating the integral of these probability multiples over the entire shaded area under that graph. Maybe an actual stats person can let us know in the comments?
Anyway, here’s the overlap graph for ranks 1 and 3: